UNIPLANAR MOTION OF A RIGID BODY 235
Therefore after the time k the hoop will roll along towards the right with a linear velocity v2 less than VQ, and with an angular velocity o> greater than OJQ.
Case III. — Suppose the initial rotation to be counter-clockwise. In this case we obtain
V3 _ s^a, (9,0
where U is the time when sliding ceases.
There are three special cases to be considered :
(a) When VQ > aa>0, vz is positive, and consequently the hoop goes on rolling towards the right.
(b) When VQ = aco0, vs = 0, and consequently at t = tz the hoop comes to rest.
(c) When VQ < oco0, v$ is negative. Therefore at the instant t = t& the hoop begins to roll backwards.
PROBLEMS.
1. Discuss the motion of the following bodies rolling down an inclined plane without slipping:
(a) A hollow cylinder of mass m and inner and outer radii TI and r2j respectively.
(b) A hoop of mass m and radius r.
(c) A sphere of mass m and radius r.
(d) A hollow sphere of mass m and inner and outer radii r^ and r2, respectively.
(e) A spherical shell of negligible thickness of mass m and of radius r.
(f) Compare the times of descent in (c) and (e).
2. A sphere is projected, without initial rotation, up a perfectly rough inclined plane. Discuss the motion.
3. A wheel which is rotating about its own axis is placed on a perfectly rough inclined plane. Discuss the motion up the plane. "